Problem: Given $ m \angle RPS = 6x - 17$, $ m \angle QPR = 9x - 31$, and $ m \angle QPS = 57$, find $m\angle QPR$. $P$ $Q$ $S$ $R$
From the diagram, we see that together ${\angle QPR}$ and ${\angle RPS}$ form ${\angle QPS}$ , so $ {m\angle QPR} + {m\angle RPS} = {m\angle QPS}$ Substitute in the expressions that were given for each measure: $ {9x - 31} + {6x - 17} = {57}$ Combine like terms: $ 15x - 48 = 57$ Add $48$ to both sides: $ 15x = 105$ Divide both sides by $15$ to find $x$ $ x = 7$ Substitute $7$ for $x$ in the expression that was given for $m\angle QPR$ $ m\angle QPR = 9({7}) - 31$ Simplify: $ {m\angle QPR = 63 - 31}$ So ${m\angle QPR = 32}$.